Coexistence of opposite opinions in a network with communities

The majority rule is applied to a topology that consists of two coupled random networks, thereby mimicking the modular structure observed in social networks. We calculate analytically the asymptotic behaviour of the model and derive a phase diagram that depends on the frequency of random opinion flips and on the inter-connectivity between the two communities. It is shown that three regimes may take place: a disordered regime, where no collective phenomena takes place; a symmetric regime, where the nodes in both communities reach the same average opinion; and an asymmetric regime, where the nodes in each community reach an opposite average opinion. The transition from the asymmetric regime to the symmetric regime is shown to be discontinuous.

[1]  H. Ohtsuki,et al.  Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs. , 2007, Physical review letters.

[2]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[3]  Grégoire Nicolis,et al.  Introduction to Nonlinear Science: References , 1995 .

[4]  C. Lee Giles,et al.  Self-Organization and Identification of Web Communities , 2002, Computer.

[5]  Stefan Bornholdt,et al.  Detecting fuzzy community structures in complex networks with a Potts model. , 2004, Physical review letters.

[6]  R. Lambiotte,et al.  Activity ageing in growing networks , 2007, physics/0701157.

[7]  M. Nowak,et al.  Evolution of indirect reciprocity , 2005, Nature.

[8]  S. Galam.  Application of statistical physics to politics , 1999, cond-mat/0004306.

[9]  Katarzyna Sznajd-Weron,et al.  Opinion evolution in closed community , 2000, cond-mat/0101130.

[10]  Guido Boella,et al.  Normative framework for normative system change , 2009, AAMAS 2009.

[11]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  V. Eguíluz,et al.  Highly clustered scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Arne Traulsen,et al.  Coevolution of strategy and structure in complex networks with dynamical linking. , 2006, Physical review letters.

[14]  R. Lambiotte,et al.  Majority model on a network with communities. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  M. Newman,et al.  Nonequilibrium phase transition in the coevolution of networks and opinions. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Serge Galam,et al.  Fashion, novelty and optimality: an application from Physics , 2005 .

[17]  M. Newman,et al.  Finding community structure in very large networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  A. Barabasi,et al.  Hierarchical Organization of Modularity in Metabolic Networks , 2002, Science.

[19]  S. Redner,et al.  Dynamics of majority rule in two-state interacting spin systems. , 2003, Physical review letters.

[20]  D. Zanette,et al.  Coevolution of agents and networks: Opinion spreading and community disconnection , 2006, cond-mat/0603295.

[21]  Petter Holme,et al.  Subnetwork hierarchies of biochemical pathways , 2002, Bioinform..

[22]  Dennis M. Wilkinson,et al.  A method for finding communities of related genes , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[23]  Hawoong Jeong,et al.  Random field Ising model and community structure in complex networks , 2005, cond-mat/0502672.

[24]  Marcel Ausloos,et al.  Self-citations, co-authorships and keywords: A new approach to scientists’ field mobility? , 2007, Scientometrics.

[25]  J. Hołyst,et al.  Phase transitions as a persistent feature of groups with leaders in models of opinion formation , 2000 .

[26]  T. Liggett.  Interacting Particle Systems , 1985 .

[27]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[28]  M E J Newman.  Assortative mixing in networks. , 2002, Physical review letters.

[29]  Francisco C. Santos,et al.  Cooperation Prevails When Individuals Adjust Their Social Ties , 2006, PLoS Comput. Biol..

[30]  H. Ohtsuki,et al.  A simple rule for the evolution of cooperation on graphs and social networks , 2006, Nature.

[31]  Fang Wu,et al.  Finding communities in linear time: a physics approach , 2003, ArXiv.

[32]  Dietrich Stauffer,et al.  Competition of languages in the presence of a barrier , 2007, physics/0702031.

[33]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[34]  Vittorio Loreto,et al.  Topology Induced Coarsening in Language Games , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Massimo Marchiori,et al.  Method to find community structures based on information centrality. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Krzysztof Suchecki,et al.  Ising model on two connected Barabasi-Albert networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  R. Lambiotte.  How does degree heterogeneity affect an order-disorder transition? , 2007 .

[38]  Jean-Pierre Eckmann,et al.  Curvature of co-links uncovers hidden thematic layers in the World Wide Web , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[39]  Didier Sornette,et al.  Endogenous Versus Exogenous Shocks in Complex Networks: An Empirical Test Using Book Sale Rankings , 2003 .

[40]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[41]  Chris Anderson,et al.  The Long Tail: Why the Future of Business is Selling Less of More , 2006 .

[42]  H E Stanley,et al.  Classes of small-world networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[43]  S. Redner,et al.  Voter model on heterogeneous graphs. , 2004, Physical review letters.

[44]  Claudio Castellano,et al.  Defining and identifying communities in networks. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[45]  M Ausloos,et al.  Uncovering collective listening habits and music genres in bipartite networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  Leon Danon,et al.  Comparing community structure identification , 2005, cond-mat/0505245.

[47]  Alessandro Vespignani,et al.  Absence of epidemic threshold in scale-free networks with degree correlations. , 2002, Physical review letters.

[48]  R. Guimerà,et al.  Functional cartography of complex metabolic networks , 2005, Nature.